#6356012

order	512
nilpotency class	3
isoclinism	isoclinic class of #6355997
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	C₂×H(5,𝔽₂)	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1683 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2,5,8)(3,9,12,16)(4,13,10,7)(6,14,15,11)(17,18)(19,24)(20,22)(21,23)(25,28)(26,29)(27,30)(31,32);(1,3,10,14,5,12,4,11)(2,6,13,16,8,15,7,9)(18,22)(23,25)(24,29)(30,31);(18,23)(21,28)(22,25)(24,30)(27,32)(29,31);(17,19)(18,24)(20,26)(21,27)(22,29)(23,30)(25,31)(28,32);(19,26)(24,29)(27,32)(30,31)

Polycyclic presentation:

2^9|a^2=h,b^2=fhi,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,g^c=gi,e^d=ei

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6263208);